Monotonicity of leaf numerical semigroups by genus

Prove that the number of leaf numerical semigroups of genus k is at most the number of leaf numerical semigroups of genus k+1 for every nonnegative integer k.

Background

Leaf numerical semigroups are the leaves of the numerical-semigroup tree. The conjecture asserts that their numbers form a nondecreasing sequence when ordered by genus.

References

If $k\in\mathbb{N}$, then $#{\mathscr{L}(\text{gen}=k)} \leq #{\mathscr{L}(\text{gen}=k+1)}$.

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Conjecture 7, Section 2